Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent (Q2809375)

The authors consider the existence of positive solutions to the singular elliptic problem NEWLINE\[NEWLINE -\Delta u=Q(x)u^{2^*-1}+\lambda u^{-\gamma} NEWLINE\]NEWLINE in a bounded, smooth domain \(\Omega\) of \(\mathbb{R}^N\) (\(N\geq 3\)) with Dirichlet boundary conditions; \(\lambda>0\) is a parameter, \(\gamma \in (0,1)\) is a constant, \(2^*=2N/(N-2)\) is the critical Sobolev exponent, and \(Q>0\) is continuous.NEWLINENEWLINEIn the case where \(Q\) is constant, and provided that \(\lambda>0\) is sufficiently small, the existence of two positive solutions has already been known by means of the method of super-solutions and variational arguments of mountain pass type. On the other hand, when \(Q\) is nonconstant, it is more difficult to deal with the loss of compactness that arises in the latter argument due to the critical exponent. Here it is assumed that \(Q\) attains its maximum at an interior point \(x_0\) such that NEWLINE\[NEWLINE Q(x)-Q(x_0)=o\big(|x-x_0|^{N-2}\big)\quad\text{as}\quad x\to x_0. NEWLINE\]NEWLINE Under the above assumptions, it is proven that the problem admits two positive solutions, provided that \(\lambda>0\) is sufficiently small. The first solution is obtained by directly minimizing the associated energy, while the second is obtained as a mountain pass. The singularity \(u^{-\gamma}\) is dealt with by an approximation procedure.